Session 07-07 - Binomial & Normal Distributions

Section 07: Probability & Statistics

Author

Dr. Nikolai Heinrichs & Dr. Tobias Vlćek

Entry Quiz - 10 Minutes

Quick Review from Session 07-06

From a contingency table, how do you calculate \(P(A|B)\)?
In a table with 200 total, 80 in category A, 60 in category B, and 30 in both. Find \(P(A|B)\).
How do you test if two variables are independent using a table?
A company has 1000 employees: 600 full-time, 400 with degrees, 280 full-time with degrees. Build the table.

Learning Objectives

What You’ll Master Today

Identify binomial experiments and their requirements
Apply the binomial formula: \(P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}\)
Calculate probabilities: “exactly k”, “at most k”, “at least k”
Use the geometric distribution for “first success” problems
Understand the normal distribution basics

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Binomial distribution problems appear on every Feststellungsprüfung!

Part A: Binomial Experiments

Requirements for Binomial

Binomial Experiment Conditions

Fixed number of trials: \(n\) is known in advance
Two outcomes: Success (probability \(p\)) or Failure (probability \(1-p\))
Independence: Trials don’t affect each other
Constant probability: \(p\) is the same for all trials

. . .

Examples: - Flipping a coin 10 times (heads = success) - Testing 50 products (defective = success) - Surveying 100 customers (satisfied = success)

The Binomial Formula

Binomial Distribution

\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]

Where: - \(n\) = number of trials - \(k\) = number of successes - \(p\) = probability of success - \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\) = number of ways

Understanding the Formula

\[P(X = k) = \underbrace{\binom{n}{k}}_{\text{arrangements}} \times \underbrace{p^k}_{\text{k successes}} \times \underbrace{(1-p)^{n-k}}_{\text{n-k failures}}\]

. . .

Example: In 5 coin flips, \(P(\text{exactly 3 heads})\)?

\[P(X=3) = \binom{5}{3} \times (0.5)^3 \times (0.5)^2 = 10 \times 0.125 \times 0.25 = 0.3125\]

Binomial Distribution Visualization

Part B: Common Probability Questions

Three Types of Questions

Question Type	Formula
Exactly k	\(P(X = k)\)
At most k	\(P(X \leq k) = \sum_{i=0}^{k} P(X=i)\)
At least k	\(P(X \geq k) = 1 - P(X < k) = 1 - P(X \leq k-1)\)

. . .

For “at least” problems, use the complement rule!

Example: Quality Control

A machine produces items with 8% defect rate. In a batch of 15 items:

\(P(\text{exactly 2 defective})\)

. . .

\[P(X=2) = \binom{15}{2} (0.08)^2 (0.92)^{13} = 105 \times 0.0064 \times 0.326 \approx 0.219\]

. . .

\(P(\text{at most 1 defective})\)

. . .

\[P(X \leq 1) = P(X=0) + P(X=1)\] \[= \binom{15}{0}(0.08)^0(0.92)^{15} + \binom{15}{1}(0.08)^1(0.92)^{14}\] \[= 0.286 + 0.373 = 0.659\]

Example Continued

\(P(\text{at least 2 defective})\)

. . .

\[P(X \geq 2) = 1 - P(X \leq 1) = 1 - 0.659 = 0.341\]

. . .

\(P(\text{between 1 and 3 defective, inclusive})\)

. . .

\[P(1 \leq X \leq 3) = P(X=1) + P(X=2) + P(X=3)\] \[\approx 0.373 + 0.219 + 0.085 = 0.677\]

Expected Value and Variance

Binomial Mean and Variance

Expected value (mean): \(\mu = E[X] = np\)
Variance: \(\sigma^2 = np(1-p)\)
Standard deviation: \(\sigma = \sqrt{np(1-p)}\)

. . .

Example: If \(n=100\) and \(p=0.3\): - Expected successes: \(\mu = 100 \times 0.3 = 30\) - Standard deviation: \(\sigma = \sqrt{100 \times 0.3 \times 0.7} = \sqrt{21} \approx 4.58\)

Break - 10 Minutes

Part C: Geometric Distribution

Waiting for First Success

Geometric Distribution

The probability that the first success occurs on trial \(n\):

\[P(X = n) = (1-p)^{n-1} \cdot p\]

Where \(p\) = probability of success on each trial.

. . .

Example: A salesperson has a 20% chance of making a sale on each call. What’s the probability the first sale is on the 4th call?

\[P(X=4) = (0.8)^3 \times 0.2 = 0.512 \times 0.2 = 0.1024\]

Geometric: Expected Trials

Expected Number of Trials

For the geometric distribution: \[E[X] = \frac{1}{p}\]

. . .

Example: If success probability is 0.25, on average how many trials until first success?

\[E[X] = \frac{1}{0.25} = 4 \text{ trials}\]

Geometric Example: Exam Style

A machine produces defective items with probability 0.05.

\(P(\text{first defective item is the 10th produced})\)

\[P(X=10) = (0.95)^9 \times 0.05 = 0.631 \times 0.05 = 0.0316\]

. . .

\(P(\text{first defective item within first 5 items})\)

\[P(X \leq 5) = 1 - P(\text{no defective in first 5}) = 1 - (0.95)^5\] \[= 1 - 0.774 = 0.226\]

Part D: Normal Distribution Basics

The Bell Curve

The 68-95-99.7 Rule

Empirical Rule

For normal distributions: - 68% of data falls within \(\mu \pm 1\sigma\) - 95% of data falls within \(\mu \pm 2\sigma\) - 99.7% of data falls within \(\mu \pm 3\sigma\)

. . .

Example: Test scores have \(\mu = 75\) and \(\sigma = 10\)

68% of students score between 65 and 85
95% of students score between 55 and 95
99.7% of students score between 45 and 105

Normal Approximation to Binomial

When \(n\) is large and \(p\) is not too extreme:

\[\text{Binomial}(n, p) \approx \text{Normal}(\mu = np, \sigma = \sqrt{np(1-p)})\]

. . .

Rule of thumb: Use when \(np \geq 5\) and \(n(1-p) \geq 5\)

. . .

Example: If \(n=100\) and \(p=0.4\): - \(\mu = 40\) - \(\sigma = \sqrt{100 \times 0.4 \times 0.6} = \sqrt{24} \approx 4.9\) - 95% of samples will have between \(40 - 9.8 = 30.2\) and \(40 + 9.8 = 49.8\) successes

Guided Practice - 20 Minutes

Practice Problem 1

A multiple choice test has 20 questions with 4 options each. A student guesses randomly on all questions.

What’s the probability of getting exactly 5 correct?
What’s the probability of getting at least 8 correct?
What’s the expected number of correct answers?
What’s the standard deviation?

Practice Problem 2 (Exam Style)

A company’s call center receives calls with 15% conversion rate.

In 10 calls, what’s the probability of exactly 2 conversions?
In 10 calls, what’s the probability of at least 1 conversion?
What’s the probability that the first conversion is on the 5th call?
On average, how many calls until the first conversion?

Wrap-Up & Key Takeaways

Today’s Essential Concepts

Binomial formula: \(P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}\)
Binomial parameters: \(\mu = np\), \(\sigma = \sqrt{np(1-p)}\)
Three question types: Exactly k, at most k, at least k
Geometric: \((1-p)^{n-1} \cdot p\) for first success on trial \(n\)
Normal: 68-95-99.7 rule for bell curves

Next Session Preview

Coming Up: Mock Exam 2

Full 180-minute exam simulation
All Section 07 topics covered
Practice under exam conditions
Prepare your formula sheet!

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Homework

Complete Tasks 07-07 - focus on binomial calculation practice!